The Inverse Problem for a Mixed Type Equation with a Fractional Order Operator in a Rectangular Domain

2021 ◽  
Vol 65 (3) ◽  
pp. 25-42
Author(s):  
B. I. Islomov ◽  
U. Sh. Ubaydullayev
Keyword(s):  
Inverse Problem ◽  
Fractional Order ◽  
Mixed Type ◽  
Type Equation ◽  
Rectangular Domain ◽  
Order Operator ◽  
Mixed Type Equation

A nonlocal inverse problem for a mixed-type equation

2011 ◽  
Vol 55 (2) ◽  
pp. 61-74 ◽  
Author(s):  
K. B. Sabitov ◽  
N. V. Martem’yanova
Keyword(s):  
Inverse Problem ◽  
Mixed Type ◽  
Type Equation ◽  
Mixed Type Equation

scholarly journals An Inverse Source Non-local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator

2017 ◽  
Vol 7 (2) ◽  
pp. 417-438 ◽  
Author(s):  
E. Karimov ◽  
N. Al-Salti ◽  
S. Kerbal
Keyword(s):  
Mixed Type ◽  
Type Equation ◽  
Integral Form ◽  
Rectangular Domain ◽  
Regularity Conditions ◽  
Mixed Type Equation ◽  
Fractional Differential Operator ◽  
Non Local ◽  
Fractional Differential ◽  
The Given

AbstractWe consider the unique solvability of an inverse-source problem with integral transmitting condition for a time-fractional mixed type equation in rectangular domain where the unknown source term depends only on the space variable. The solution is based on a series expansion using a bi-orthogonal basis in space, corresponding to a non-self-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. The influence of the transmitting condition on the solvability of the problem is also demonstrated. Two different transmitting conditions are considered — viz. a full integral form and a special case. In order to simplify the bulky expressions appearing in the proof of our main result, we establish a new property of the recently introduced Mittag-Leffler type function in two variables.

THE BOUNDARY VALUE PROBLEM OF FINDING A RIGHT-HAND SIDE FOR MIXED TYPE EQUATION WITH CHAPLYGIN’S TYPE OPERATOR

2021 ◽  
Vol 9 (4) ◽  
pp. 26-50
Author(s):  
K.B. Sabitov ◽  
◽  
I.A. Burkhanova (Haji) ◽  
Keyword(s):  
Inverse Problem ◽  
Mixed Type ◽  
Orthogonal Series ◽  
Type Equation ◽  
Transition Line ◽  
Mixed Type Equation ◽  
Uniqueness Criterion ◽  
Right Hand ◽  
Definition Of ◽  
The Given

In this paper, we study the inverse problem for a mixed-type equation with power degeneracy on a transition line by definition of its right-hand side, depending on the spatial coordinate. The theory of identity has been proved. In the case of degree degeneracy, the uniqueness criterion for the solution of the problem is proved, and the solution itself is con- structed in the form of a sum of orthogonal series. The consistency of series in the class of solutions of the given equation is justified and the validity of the solution with respect to the boundary conditions is proved.

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